Reservoir pressure and temperature changes associated with oilfield operations can modify the in situ stress field, causing deformation (induced strains) within the stressed rock units. Such stress-strain response can dramatically impact petroleum engineering practices encompassing drilling, completing and stimulating wells to full-field reservoir recovery and optimization. To this end, the science of geomechanics seeks to understand and predict the impact of rock deformation (stress-strain behavior) in petroleum engineering applications, using numerical models of the subsurface.
All geomechanical models on the single-well to full-field scale, whether simple analytical calculations or complex fully-coupled finite-element simulations, require some knowledge of mechanical properties in order to describe the deformation behavior of the stressed rock units. However, mechanical properties can only be directly quantified from laboratory testing of recovered core from the lithologic units of interest. While much effort has been expended in designing sophisticated experimental techniques for measuring rock mechanical properties, subsurface model results will be severely limited if no core is available from the specific rock strata. Further, in many cases core measurements will be highly limited and thus discontinuous, biased towards mechanically stronger intervals due to core recovery issues and almost always absent from non-reservoir overburden and underburden lithologies.
Due to the impracticalities often associated with obtaining rock core for laboratory measurement of mechanical properties, the ability to predict such properties from other more easily measured or remotely-sensed material parameters has received widespread attention within the petroleum, civil and mining engineering, geophysical and structural geology communities over the last approximately fifty years.
Diverse analytical schemes exist that attempt to predict “elastic” mechanical properties (describing small-strain rock deformation behavior) through consideration of the influence of microstructure (grains, pores and cracks) on macroscopic constitutive behavior. For example, Mavko (1998), Zimmerman (1991) and Hill (1952) offer techniques for estimating the effective elastic (bulk and shear) moduli of a solid rock from knowledge of the volume fractions and moduli of the various mineral components. Numerous effective medium theories also exist that allow estimation of the effective elastic moduli of porous and cracked rocks (e.g. O'Connell and Budiansky, 1974, Kuster and Toksoz, 1974) including the impact of pore structure on elastic compressibility (Zimmerman, 1986) and elastic nonlinearity resulting from crack closure (Zimmerman, 1991). While estimation of elastic rock properties has practical geomechanical application in, for example, the modeling of hydraulic fracture containment (e.g. Simonson et al., 1978) knowledge of effective elastic moduli alone is insufficient to describe “plastic” (large-strain deformation) in which stresses approach the failure strength of the rock.
Accordingly, several micromechanical models have also been developed that attempt to predict such large-strain constitutive behavior, specifically with reference to the onset of brittle failure under compressive loading. Griffith (1924) with modifications by McClintock & Walsh (1962) and Murrell (1963) utilized a theoretical rock model containing a distribution of randomly orientated thin elliptical cracks to derive a failure criterion capable of qualitatively describing the dependence of rock strength on stress state. More recently, a number of stochastic/statistical modeling approaches (summarized in Yuan & Harrison 2006) have been developed for simulating the process of progressive mechanical breakdown of initially intact heterogeneous rocks. However, while the above micromechanical models are selectively capable of qualitatively reproducing many of the phenomenological features associated with elastic-brittle rock deformation both pre- and post-failure (e.g. onset of the brittle-ductile transition, relationships between localized damage and hydraulic flow) collectively they are impractical for the purposes of predicting large-strain plastic rock mechanical properties with sufficient accuracy for petroleum geomechanics modeling purposes. Additional limitations might also include: (i) computational intensity; (ii) lack of sufficient petrographic characterization of required microstructural input parameters; (iii) necessity of core measurements for model calibration purposes; (iv) difficulties associated with upscaling results.
As a direct result of the above perceived deficiencies in theoretical models of large-strain plastic behavior (by comparison with those models developed for effective elastic moduli estimation) most effort has concentrated on generating practical empirical relationships for inferring mechanical properties from knowledge of other more easily measured rock properties. For example, rock strength is frequently inferred from other petrophysical properties routinely derived from geophysical wireline logs, despite the fact that no logs directly measure the actual strength of the logged formation. The rationale behind such predictive relationships is that the same petrographic parameters that notionally control rock strength (such as porosity, mineralogy, clay content, cement content, grain size etc.) also impact other log-derived petrophysical parameters (such as acoustic velocities and dynamic elastic moduli) in a similar manner. That is, the same combination of petrographic parameters that make a particular lithology relatively more compliant (less stiff) and thus slower with respect to the speed of acoustic wave propagation, also decrease its strength (load-bearing capacity) relative to surrounding lithologies.
A plethora of such empirical strength—petrophysical property correlations exist in the literature and are well summarized in Chang et al. (2006). Nearly all proposed formulae utilize a combination of one or more of the following parameters to predict rock strength: (i) P-wave or S-wave acoustic velocities (or the equivalent interval transit times which are directly measured via acoustic logging tools or on core plugs in the laboratory); (ii) dynamic elastic constants (Young's modulus or Poisson's ratio) derived from acoustic velocities (P- & S-waves) and bulk density; (iii) porosity directly measured from core plugs, or calculated from sonic, density, neutron or magnetic resonance logs. While some empirical models work reasonably well for a particular geographic region or specific lithology, considerable scatter in predicted strength indicates that none are sufficiently generic to fit all available data. Also, most current empirical strength—petrophysical property correlations predict only the unconfined compressive strength (rock strength under atmospheric pressure conditions) and therefore ignore the strengthening effect associated with minimum principal stresses elevated above atmospheric pressure conditions.
It is believed that there are no existing techniques that can predict a range of plastic mechanical properties in a diversity of lithologies without recourse to obtaining at least one core sample from the strata of interest for laboratory testing. However, there are multiple relevant technologies for general mechanical properties prediction. A brief review and summary of the limitations of two such technologies for rock strength and compressibility prediction follows next.
Rock Strength Prediction
The mechanical strength of sedimentary rocks is fundamental data input to many geomechanical analyses applied in the petroleum industry. Accurate estimation of compressive rock strength is essential for, amongst others: (i) drilling simulation of penetration rate and drill bit wear (Cooper and Hatherly 2003); (ii) wellbore stability analysis (McLean and Addis 1990); (iii) sand production prediction (Nouri et al., 2006); (iv) constraining in situ stress magnitudes from observations of wellbore failure (Peska and Zoback, 1995).
Deere and Miller (1966) developed a mechanical properties database for a wide range of igneous, metamorphic and sedimentary rocks. Once the data are subdivided on the basis of lithology, individual correlations between static Young's modulus and unconfined compressive strength become apparent for broad rock type classes including sandstones, limestones and shales. Coates and Denoo (1981) utilized these trends to predict formation strength from wireline log data in order to estimate well production rates resulting in formation collapse and sand production. Using Deere and Miller's sandstone and shale lines, Coates and Denoo developed an empirical relationship for estimating unconfined compressive strength from knowledge of Young's modulus (derived from bulk density, compressional and shear acoustic log data) and the fraction of clay in the sandstone (derivable from conventional open-hole log analysis). Mohr-Coulomb cohesive strength is then obtained from the unconfined compressive strength by multiplying by the bulk modulus (again derived from log data). In an analysis of the above technique, Fjaer et al. (1992) show convincingly that the above derivation for cohesive strength is primarily dependent on compressional velocity, such that a 10% change in velocity results in a 46% change in strength estimate. As noted by Fjaer, significant limitations associated with the above technique include: (i) sizeable differences between static (core-derived) and dynamic (log-derived) elastic moduli can result in considerable error when using static correlations to predict strength from elastic moduli calculated from sonic velocities; (ii) as compressional velocity is dependent on applied stress and fluid saturation in addition to rock microstructure, strength predicted using the above technique should be field dependent, that is should vary with subsurface depth and reservoir environment for a constant lithology; (iii) since the lowest strength in the mechanical properties database is ˜30 MPa such that the data represent relatively high strength rocks, it is unclear how the empirical correlation will extrapolate to low strength formations; (iv) a constant Mohr-Coulomb friction angle of 30 degrees is assumed in the above analysis, whereas laboratory measured friction angles in equivalent sandstone-to-shale lithologies is known to vary from ˜5 degrees to ˜50 degrees.
Raaen et al. (1996) summarize a proprietary technique that utilizes a constitutive model describing micromechanical deformation processes, to derive large-strain rock mechanical properties such as rock strength from dynamic (low-amplitude strain) acoustic data. Conceptually the empirical formula relies on quantifying the micromechanisms that give rise to differences between the static and dynamic elastic moduli, such as grain contact crushing, pore collapse and microcrack shear slippage. Raaen et al. contend that, while differences between static and dynamic moduli are partly attributable to fluid effects, the above plastic micromechanisms requiring large strain amplitude for activation account for the majority of static versus dynamic mismatch, as small amplitude dynamic loading resulting from acoustic excitation cannot trigger such mechanisms. The model relies on two sets of input parameters—one set describing micromechanical processes responsible for static versus dynamic mismatch are derived from laboratory measurements and are field independent. The other field-specific set includes formation volumetrics (sand-shale-oil-water) compressional and shear acoustic logs and in situ stress estimates. As the main focus of this technique is to provide sand strength estimation for quantifying solids production in flowing wells, an obvious limitation of the model is an inability to predict the strength of argillaceous, shaly rocks which can often contribute to wellbore stability issues.
Smith and Goldman (1998) describe a patented technique for assaying the compressive strength of rock, primarily based upon a database of laboratory measurements from which strength versus porosity correlations are extracted. While the technique mainly addresses unconfined compressive strength, it can potentially accommodate other factors that impact rock strength, such as confining stress, bedding plane dip and temperature. Due to the predictive methodology underlying the technique, ideally calibration involving laboratory testing of “pure lithologies” (clean sandstone and/or shale) recovered from the well of interest is required, however lithologically similar samples (perhaps from offset wells) would suffice. The technique has limitations when used to predict strength in “mixed lithologies” such as shaly sandstones or sandy shales, as it relies on a weighted average of the pure end member components which can lead to significant predictive uncertainty.
Birchwood (2008) details a patented technique for managing a drilling operation in a multicomponent particulate system (specifically a sediment-hydrate system). The technique involves obtaining petrophysical, in situ stress and geophysical properties of a sedimentary system of interest, constructing correlations between these properties and various elastoplastic properties of the sedimentary system (e.g. dilation angle, unconfined compressive strength, cohesion hardening parameters etc.) based on micromechanical models governing the deformation of granular materials, and using these correlations to estimate the elastoplastic properties of the sedimentary system of interest. Limitations associated with this method are as follows: (i) the method involves obtaining measurements of one or more static drained elastoplastic properties from the sedimentary system of interest (ii) the micromechanical model involves diverse microstructural parameters (such as particle coordination number, grain-size and grain-to-grain friction coefficients) which are difficult to characterize without sampling the strata of interest; (iii) the micromechanical model may not be applicable to diverse lithologies such as carbonates and shales.
Rock Compressibility Prediction
The compressibility of reservoir formations (specifically the uniaxial pore volume compressibility, or “UPVC”) is a fundamental mechanical rock property utilized in many reservoir engineering calculations including reserves estimates, reservoir performance and production forecasting. For example, ignoring UPVC in material balance calculations can result in over estimation of oil-in-place and prediction of excessive volumes of water influx (Hall, 1953). In undersaturated reservoirs (fluid pressures in excess of the bubblepoint), liquid and formation (UPVC) compressibilities represent significant proportions of the total compressibility of the reservoir system and therefore strongly impact the prediction of reservoir performance. As fluid properties are generally well constrained, uncertainty in UPVC directly contributes to inaccuracies in material balance and simulation results (Carlson, 2003). Pressure decline above the bubblepoint can be very rapid, such that a substantial amount of recovery can occur prior to obtaining core and measuring mechanical rock properties. Of equal importance is the uniaxial bulk volume compressibility, or “UBVC,” which is used in reservoir compaction and surface subsidence calculations and for well operability limit modeling (e.g. Guenther et al., 2005). Rock (both UPVC and UBVC) compressibilities can be accurately quantified only via geomechanical measurements on recovered core in which care is taken to simulate in situ stress magnitudes and boundary conditions (e.g. Lachance and Andersen, 1983; Rhett and Teufel, 1992). To accommodate this need for data early in the production lifecycle, a technique for predicting rock compressibility from geophysical wireline logs offers considerable advantage. Further, discrete core measurements are rarely obtained in sufficient number to overcome between-sample data variability.
In the petroleum industry there are two basic laboratory techniques employed to determine pore compressibility magnitude: (i) hydrostatic compression giving HPVC; (ii) uniaxial strain boundary conditions giving UPVC. While hydrostatic testing is much simpler to perform in the laboratory, in general in situ reservoir compaction is better approximated by uniaxial strain loading (one-dimensional vertical compaction with zero expansion or contraction in the radial direction). Theoretically, HPVC is related to UPVC through a “uniaxial strain correction factor, USCF”. For elastic (small strain, recoverable) deformation, Teeuw (1981) showed that USCF is a function of Poisson's ratio, v. For plastic (large strain, irrecoverable) deformation, it can be shown that USCF is a function of the ratio of horizontal-to-vertical effective stress change, kO. In the absence of geomechanical measurements of UPVC on recovered core, normal oilfield practice is to utilize pore compressibility HPVC versus porosity correlations derived from extensive databases of hydrostatic core measurements. For example Newman (1973) provided correlations for siliciclastic reservoir rocks, subdivided into consolidated, friable and unconsolidated sand lithotypes.
FIG. 1 is a plot of uniaxial pore volume compressibility, UPVC, as a function of fractional porosity, φ (both measured at initial in situ reservoir stress conditions prior to production) derived from ˜three hundred uniaxial strain tests performed by the present inventors on siliciclastic reservoir rocks. It is evident from the data point scatter that UPVC and porosity are not highly correlated. Newman's HPVC predictors (the curves in the drawing) for unconsolidated, friable and consolidated sands are superimposed on the uniaxial core data with a range of admissible elastic and plastic USCF's. A limitation of this technique is apparent from FIG. 1 in that, without knowing the generic lithotype (consolidated versus friable versus unconsolidated sand) and the appropriate USCF, porosity does not yield a reliable prediction of UPVC. Accordingly material balance and simulation results above the bubblepoint could not be considered accurate if formation compressibility is based upon such correlations as opposed to core measurements.
Khatchikian (1995) outlines a methodology for deriving UPVC directly from geophysical wireline logs. Dynamic “wet” uniaxial bulk compressibility, UBVC is first calculated from the compressional and shear slowness and the bulk density logs using standard dynamic moduli formulations (e.g. Mavko et al., 1998). Dynamic ‘wet” UBVC is then converted to an equivalent static “dry” value by removing fluid stiffness effects through low frequency Biot-Gassmann substitution (e.g. Mavko et al., 1998). Finally, UPVC is computed from the log-derived “dry” UBVC, the matrix compressibility and the porosity according to relationships given by Zimmermann (1991). While Khatchikian's methodology is valid for estimating essentially elastic UPVC at reservoir stress conditions (as sonic logs measure formation stiffness resulting from low-amplitude, high-frequency elastic deformation) it is incapable of predicting compressibility associated with any significant component of large-strain plastic deformation.
Ong et al. (2001) and Wolfe et al. (2005) describe the same wireline log-based methodology for determining UPVC. For input, the method requires static mechanical properties that characterize the rock strata of interest. Both Ong and Wolfe use the previously described micromechanical constitutive model of Raaen et al. (1996) to generate the equivalent static mechanical properties from dynamic log data. From previously established relationships between porosity, bulk density, mineralogy, dynamic properties and grain contact, sliding crack and dilatancy parameters, the micromechanical model uses fluid saturations, lithology, porosity, density and shear and compressional slowness to construct a “virtual core sample.” Knowledge of the in situ stress and pore pressure fields are required to ascertain the initial confining conditions to apply to each virtual core sample. The constitutive behavior (and thus static mechanical properties) of each virtual core sample representing the strata of interest is then examined through simulated hydrostatic and triaxial loading. Incremental stresses and strains are calculated and static loading stress-strain curves constructed. Static mechanical properties including rock strength, elastic moduli and compressibilities are then deduced from the simulated stress-strain curves. Finally, bulk compressibility measured over the linear elastic portion of the simulated triaxial deviatoric stress-strain curve (between 5% and 75% of peak strength) is converted to pore compressibility using Zimmerman's (1991) transforms, which in turn is converted to an equivalent UPVC using the theoretical elastic solution of Teeuw (1981). Thus uniaxial strain pore volume compressibility is derived from a micromechanical constitutive model used to simulate triaxial compression of a virtual core specimen representing the strata of interest. Limitations of this technique are as follows: (i) as the method requires simulated loading of virtual core samples representing each strata of interest, it is impractical for wells penetrating lithologically diverse strata; (ii) propagation of error associated with micromechanical modeling of static mechanical properties could result in significant error in derived UPVC's; (iii) ideally the micromechanical model should be calibrated against laboratory testing of recovered core; and (iv) knowledge of the in situ stress and pore pressure fields may not be practical, particularly in an exploration setting.